Velocity obstacles [1]} was proposed in 2008 as a new concept for real-time multi agent navigation. Let \(A\)  and \(B\)  be two vessels positioned at \(P_{a}\)  and \(P_{b}\)  and moving in the direction of their arrows with velocities of \(V_{a}\)  and \(V_{b}\)  as shown in Figure REF . We assume that \(A\)  is an ASV and \(B\)  is a human controlled vessel. The velocity obstacle \(VO_{b}^{a}(\vec{V_{b}})\)  of vessel \(B\)  to the ASV is the set consisting of all those velocities \(V_{a}\)  for ASV that will result in a collision at some moment in time with vessel \(B\)  moving at velocity \(V_{b}\) . Figure REF  shows the velocity obstacle \(VO_{b}^{a}(\vec{V_{b}})\)  of a disc-shaped vessel \(B\)  to a disc-shaped ASV. For simplicity we consider disc-shaped vessels where the radius of the enclosing disc is the length of the vessel. In order to avoid collision, two vessels' representative discs must not touch each other at any given time. Let \(A \oplus B \)  denote the Minkowski sum of vessels \(A\)  and \(B\)  and let\(-A\)  denote the vessel \(A\)  reflected in its reference point.
\(A \oplus B = \lbrace a + b | a \in A, b \in B \rbrace , -A = \lbrace -a | a \in A\rbrace \) 
